Len Netcott, chairman of the Clifton and Hotwells branch, said he was told on Tuesday a tin had been stolen from the Post Office on Gloucester Road.
"In an hour I was getting more calls to say the hardware store and a florist had them stolen as well," he said.
Avon and Somerset Police said it would be investigating the thefts.
"Unfortunately the Post Office didn't have any evidence, but the Pearce's Hardware Store had CCTV of the theft," Mr Netcott said.
"The guy walked right into the camera with his face on full view.
"I would like anyone who knows this chap to contact the police and see that the man gets his just desserts because he is stealing from people less fortunate than himself."
'Despicable crime'
An Avon and Somerset Police spokeswoman said: "I can assure you that investigations into three separate thefts in the Horfield and Bishopston area are very much ongoing.
"We take any reports of theft very seriously, however theft of charity tins is particularly a despicable crime.
"These kinds of thefts are particularly cruel because the money stolen from the tins would otherwise be donated to charity and therefore affects those that need the money most.
"I would urge anyone who knows who may be responsible for these thefts to contact us," she added.
